Average Error: 14.7 → 12.2
Time: 6.2s
Precision: 64
\[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.4801961206040853 \cdot 10^{-87} \lor \neg \left(a \le -1.46265163071345889 \cdot 10^{-196} \lor \neg \left(a \le 3.0921765523630852 \cdot 10^{-306} \lor \neg \left(a \le 2.3190386960694568 \cdot 10^{-9}\right)\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \left(\sqrt[3]{y - z} \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right), \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \end{array}\]
x + \left(y - z\right) \cdot \frac{t - x}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.4801961206040853 \cdot 10^{-87} \lor \neg \left(a \le -1.46265163071345889 \cdot 10^{-196} \lor \neg \left(a \le 3.0921765523630852 \cdot 10^{-306} \lor \neg \left(a \le 2.3190386960694568 \cdot 10^{-9}\right)\right)\right):\\
\;\;\;\;\mathsf{fma}\left(\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \left(\sqrt[3]{y - z} \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right), \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r130869 = x;
        double r130870 = y;
        double r130871 = z;
        double r130872 = r130870 - r130871;
        double r130873 = t;
        double r130874 = r130873 - r130869;
        double r130875 = a;
        double r130876 = r130875 - r130871;
        double r130877 = r130874 / r130876;
        double r130878 = r130872 * r130877;
        double r130879 = r130869 + r130878;
        return r130879;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r130880 = a;
        double r130881 = -1.4801961206040853e-87;
        bool r130882 = r130880 <= r130881;
        double r130883 = -1.462651630713459e-196;
        bool r130884 = r130880 <= r130883;
        double r130885 = 3.092176552363085e-306;
        bool r130886 = r130880 <= r130885;
        double r130887 = 2.319038696069457e-09;
        bool r130888 = r130880 <= r130887;
        double r130889 = !r130888;
        bool r130890 = r130886 || r130889;
        double r130891 = !r130890;
        bool r130892 = r130884 || r130891;
        double r130893 = !r130892;
        bool r130894 = r130882 || r130893;
        double r130895 = y;
        double r130896 = z;
        double r130897 = r130895 - r130896;
        double r130898 = cbrt(r130897);
        double r130899 = r130898 * r130898;
        double r130900 = t;
        double r130901 = x;
        double r130902 = r130900 - r130901;
        double r130903 = cbrt(r130902);
        double r130904 = r130903 * r130903;
        double r130905 = r130880 - r130896;
        double r130906 = cbrt(r130905);
        double r130907 = r130906 * r130906;
        double r130908 = r130904 / r130907;
        double r130909 = r130898 * r130908;
        double r130910 = r130899 * r130909;
        double r130911 = r130903 / r130906;
        double r130912 = fma(r130910, r130911, r130901);
        double r130913 = r130901 / r130896;
        double r130914 = r130900 / r130896;
        double r130915 = r130913 - r130914;
        double r130916 = fma(r130895, r130915, r130900);
        double r130917 = r130894 ? r130912 : r130916;
        return r130917;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Derivation

  1. Split input into 2 regimes

  2. if a < -1.4801961206040853e-87 or -1.462651630713459e-196 < a < 3.092176552363085e-306 or 2.319038696069457e-09 < a

    1. Initial program 11.1

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]

    2. Simplified11.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)}\]
    3. Using strategy rm

    4. Applied fma-udef11.1

      \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt11.5

      \[\leadsto \left(y - z\right) \cdot \frac{t - x}{\color{blue}{\left(\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}\right) \cdot \sqrt[3]{a - z}}} + x\]

    7. Applied add-cube-cbrt11.7

      \[\leadsto \left(y - z\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}\right) \cdot \sqrt[3]{t - x}}}{\left(\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}\right) \cdot \sqrt[3]{a - z}} + x\]

    8. Applied times-frac11.7

      \[\leadsto \left(y - z\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\right)} + x\]

    9. Applied associate-*r*9.4

      \[\leadsto \color{blue}{\left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}} + x\]
    10. Using strategy rm

    11. Applied fma-def9.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}, \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}, x\right)}\]
    12. Using strategy rm

    13. Applied add-cube-cbrt9.4

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \sqrt[3]{y - z}\right)} \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}, \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}, x\right)\]

    14. Applied associate-*l*9.4

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \left(\sqrt[3]{y - z} \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right)}, \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}, x\right)\]

    if -1.4801961206040853e-87 < a < -1.462651630713459e-196 or 3.092176552363085e-306 < a < 2.319038696069457e-09

    1. Initial program 23.1

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]

    2. Simplified23.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)}\]

    3. Taylor expanded around inf 21.9

      \[\leadsto \color{blue}{\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}}\]

    4. Simplified18.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.4801961206040853 \cdot 10^{-87} \lor \neg \left(a \le -1.46265163071345889 \cdot 10^{-196} \lor \neg \left(a \le 3.0921765523630852 \cdot 10^{-306} \lor \neg \left(a \le 2.3190386960694568 \cdot 10^{-9}\right)\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \left(\sqrt[3]{y - z} \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right), \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  :precision binary64
  (+ x (* (- y z) (/ (- t x) (- a z)))))